Continuous Displacement of “Lattice” Atoms

In the existing CVM (Cluster Variation Method) formulations, atoms are placed on lattice points. A modification is proposed in which an atom can be displaced from a lattice point. The displaced position is written by a vector r, which varies continuously. This model is treated in the CVM framework by regarding an atom at r as a species r. The probability of finding an atom displaced at r in dr is written as f(r)dr, and the corresponding pair probability is written as g(r1, r2)dr1dr2. We formulate using the pair approximation of the CVM in the present paper. The interatomic potential is assumed given, for example as the Lennard-Jones form. The entropy is written in terms of f(r) and g(r1, r2) using the CVM formula. The special feature of the present formulation which is different from the prevailing no-displacement cases of the CVM is that rotational symmetry of the lattice is to be satisfied by the f(r) and g(r1, r2) functions. After the general equations are written in the continuum vector form and in the integral equation formulation, an example of a single-component system is solved by changing integrals into summations over finite intervals. Further we construct simulations of displacement patters in such a way that the pattern satisfies the pair probability distribution which has been calculated as the output of the CVM analysis. The simulated pattern shows the wavy behavior of phonons. Future directions are discussed.